The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < \omega \rangle$ of models of size at most $\aleph_1$ in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter $\mathcal D$ on $\omega$, if the ultraproducts $\prod\limits_{\mathcal D} \mathbb{M}^{1}_n$ and $\prod\limits_{\mathcal D} \mathbb{M}^{2}_n$ are elementarily equivalent, then they are isomorphic.